6.11. Sample size determination
The reason why we always conduct a sample observations and not a census is that almost always we have limited resources at our disposal. In our calculations, if a smaller sample can serve our purpose, then we will be wasting our resources by taking a larger sample. For example, suppose we want to estimate the mean life of certain type of lights bulbs. If a sample of 50 light bulbs can give us the type of confidence interval that we are looking for, then we will be wasting money and time if we take a sample of much larger size, say 800 light bulbs. In such cases if we know the confidence interval that we want, then we can find the (approximate) size of the sample that will produce the required result.
6.11.1. Sample size determination for the estimation of mean
Suppose that sample of n observations is taken from a normally distributed population with mean and known variance . We know that confidence interval for the population mean is given by
where is the sample mean and is the appropriate cutoff point of the standard normal distribution. This confidence interval is centered on the sample mean and extends a distance of L, the margin of error (also called the sample error, the bound, or the interval half width) is given by
Suppose that we predetermine the size of L and want to find the size of the sample that will yield this margin error. From the above expression, the following formula is obtained that determines the required sample size n.
Definition:
Given the confidence level and standard deviation of the population
(or population variance), the sample size that will produce a predetermined margin error L of the confidence interval estimate of is
Remark 1:
If we do not know , we can take a sample and find sample standard deviation. Then we can use S for in the formula.
Remark 2:
n must be rounded to the next higher integer, because a sample size can not be fractional.
Example:
Suppose that we want to estimate the mean family size for all country families at 99 % confidence level. It is known that the standard deviation
for the sizes of all families in the country is 0.45.
How large a sample should we select if we want its estimate to be within 0.02 of the population mean?
Solution:
We want the 99 % confidence interval for the mean family size to be
.
Hence, the margin of errors is to be 0.02, that is
The value of for a 99 % confidence level is 2.58.
The value of is given to be 0.45. Therefore, substituting all values in the formula and simplifying, we obtain
Thus, the required sample size is 3370. If we will take a sample of 3370 families, compute the mean family size for this sample, and then margin of a 99 % confidence interval around this sample, the margin of error of the estimate will be approximately 0.02.
6.11.2. Sample size determination for the estimation of proportion
Just as we did with the mean, we can also determine the sample size for estimating the population proportion p.
We know that confidence interval for p is given by
where - is the sample proportion.
This interval is centered on the sample proportion and extends a distance L:
This
result can not be used directly to determine the sample size n
necessary to obtain a confidence interval of some specific width,
since it involves
,
which is not known. But whatever the outcome,
can not be bigger than 0.25, its value when the sample proportion is
0.5. Thus, the largest possible value for L
is given by
Using basic algebra, we obtain
and squaring yields
Definition:
Let a random sample be selected from a normal population.
confidence interval for the population proportion, extending a distance of at most L on each side of the sample proportion, can be guaranteed if the sample size is
Example:
A public health survey is to be designed to estimate the proportion p of a population having defective vision. How many persons should be examined if the public health doctor wishes to be 98 % certain that error of estimation is below 0.05?
Solution:
The
public health doctor wants the 98 % confidence interval to be
Therefore
.
The value of
for
a 98 % confidence level is 2.33.
After substituting we obtain that the required sample size is
.
Thus, if the doctor takes a sample of 543 persons, the estimate of p will be within 0.05 of the population proportion.
Exercises
1. Determine the sample size for the estimate of for the following:
a)
; 
; confidence
level = 99%
b)
; 
; confidence
level = 95%
c)
; 
; confidence
level = 90%
2. Determine the most conservative sample size for estimation of the population proportion for the following:
a)
; confidence
level = 99 %
b)
; confidence
level = 96 %
c)
; confidence
level = 90 %
3.
A sample of 50 workers’ average weekly earnings gave
.
Determine the sample size that is needed for estimating the
population mean weekly earnings with a 98 % error margin of $ 3.50.
4. How large a sample should be taken to be 95 % sure that the error of estimation does not exceed 0.02 when estimating a population proportion?
5.
A food service manager wants to be 95 % confident that the error in
the estimate of the mean number of sandwiches dispensed over the
lunch hour is 10 or less. What sample size should be selected if
6. One department manager wants to estimate at 90 % confidence level the mean amount spent by all customers at this store. He knows that the standard deviation of amounts spent by customers at this store is $ 27. What sample size he chooses so that the estimate is within $ 3 of the population mean?
7. A teacher wants to estimate the proportion of all students who own mobile telephones. How large should the sample size be so that the
99 % confidence interval for the population proportion has a maximum
error of 0.03?
8. A private university wants to determine a 99 % confidence interval for the mean number of hours that students spend per week doing homework. How large a sample should be selected so that the estimate is within 1 hour of the population mean? Assume that the standard deviation for the time spent per week doing homework by students is 3 hours.
Answers
1. a) 186; b) 126; c) 61; 2. a) 2653; b) 437; c) 3007; 3. 543; 4. 2401; 5. 62;
6. 220; 7. 1842; 8. 60.